Hi folks,
I am trying to understand Bayesian neural networks and have come across several mathematical equations that I have not been able to follow. I would like to understand the rules of calculation to compute an integral over a product of two Gaussian distributions. What I know is that the product of two Gaussian is again a Gaussian, so it would be enough for me to have a calculation rule how to calculate the mean and the variance of this distribution.
First example - Two distributions over same random variable:
[MATH]\int N(a; 0, \frac{1}{\lambda^2}) \cdot (a; \mu_a, \sigma^2_a) \ da[/MATH]
Second example:
[MATH]\int _\mathbb{R} p(\underline{w}|a, D_i)\cdot p(a|D_i) \ da[/MATH]
with
[MATH]p(\underline{w}|a, D_i) = N\left(\underline{w};\underline{u}^w_{i-1}+\underline{l}_i \cdot (a-\mu_a), \mathbf{C}^w_{i-1}-\underline{l}_i \cdot \underline{\sigma}_{wa} ^T\right)[/MATH] where [MATH]\underline{l}_i=\underline{\sigma}_{wa}/\sigma^2_a[/MATH] and [MATH]\underline{\sigma}_{wa}=\mathbf{C}^w_{i-1}\cdot \underline{x}_i[/MATH]
and
[MATH]p(a|D_i) = N(a;\mu_i,\sigma^2_i)[/MATH]
Are there any general rules that can be applied to these two integrals in order to solve them in closed form? Thanks for any hint!
I am trying to understand Bayesian neural networks and have come across several mathematical equations that I have not been able to follow. I would like to understand the rules of calculation to compute an integral over a product of two Gaussian distributions. What I know is that the product of two Gaussian is again a Gaussian, so it would be enough for me to have a calculation rule how to calculate the mean and the variance of this distribution.
First example - Two distributions over same random variable:
[MATH]\int N(a; 0, \frac{1}{\lambda^2}) \cdot (a; \mu_a, \sigma^2_a) \ da[/MATH]
Second example:
[MATH]\int _\mathbb{R} p(\underline{w}|a, D_i)\cdot p(a|D_i) \ da[/MATH]
with
[MATH]p(\underline{w}|a, D_i) = N\left(\underline{w};\underline{u}^w_{i-1}+\underline{l}_i \cdot (a-\mu_a), \mathbf{C}^w_{i-1}-\underline{l}_i \cdot \underline{\sigma}_{wa} ^T\right)[/MATH] where [MATH]\underline{l}_i=\underline{\sigma}_{wa}/\sigma^2_a[/MATH] and [MATH]\underline{\sigma}_{wa}=\mathbf{C}^w_{i-1}\cdot \underline{x}_i[/MATH]
and
[MATH]p(a|D_i) = N(a;\mu_i,\sigma^2_i)[/MATH]
Are there any general rules that can be applied to these two integrals in order to solve them in closed form? Thanks for any hint!